Question: All of the 5th grade teachers and students from Almond went on a field trip to an art museum. Tickets were $$8.50$ each for teachers and $$4.00$ each for students, and the group paid $$53.00$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$25.50$ each for teachers and $$8.50$ each for students, and the group paid $$127.50$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+4y = 53}$ ${25.5x+8.5y = 127.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-25.5x-12y = -159}$ ${25.5x+8.5y = 127.5}$ Add the top and bottom equations together. $ -3.5y = -31.5 $ $ y = \dfrac{-31.5}{-3.5}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {8.5x+4y = 53}$ to find $x$ ${8.5x + 4}{(9)}{= 53}$ $8.5x+36 = 53$ $8.5x = 17$ $x = \dfrac{17}{8.5}$ ${x = 2}$ You can also plug ${y = 9}$ into $ {25.5x+8.5y = 127.5}$ and get the same answer for $x$ ${25.5x + 8.5}{(9)}{= 127.5}$ ${x = 2}$ There were $2$ teachers and $9$ students on the field trips.